Mercator projection

The Mercator projection used in cartography is a named after Gerhard Mercator shape of the cylindrical projection in which the projection is distorted suitable in the direction of the cylinder axis to achieve a conformal mapping of the earth's surface. The Mercator projection is any projection in physical terms, and therefore can not be geometrically constructed. The conformality is that geometric shapes in miniature remain synonymous with conformity, undistorted. In contrast, the Mercator projection is neither equal area nor faithful direction, ie surface areas have at various points of the figure different scales, and great circles as the shortest connections between two points are not presented as straight lines. Length loyalty exists only along one or two excellent lines. After the position of the axis of the projection cylinder to the Earth, a distinction the normal Mercator projection, in which the two axes are identical, transverse Mercator projections with cylinder axes perpendicular to the Earth's axis and oblique Mercator projections in the other cases.

Mercator projections are used in particular in surveying and navigation application in surveying predominantly as transverse projections with different axes for different meridian (UTM, Gauss -Krüger projection, etc.) in the navigation as a normal projection. Named is the Mercator projection for the mapmaker Mercator, who published for navigation purposes in 1569 a card of this type in the normal position of the imaging surface on which a steered course could be plotted as a straight line for the first time.

Dissemination

The Earth's surface is in first approximation a spherical surface, which can not be mapped without distortion on a flat map. On the Internet, both free projects like OpenStreetMap and commercial providers such as Google Maps, Bing Maps and Yahoo Maps for two-dimensional representations preferably use this projection.

In surveying the transverse Mercator projection ( Gauss -Krüger projection or UTM coordinate system ) is used instead of the normal Mercator projection most.

Construction

To the earth, an imaginary cylinder is placed, which they touched along a great circle, or cut into two circles on either side of this great circle. Of the cylinder axis from any point of the globe can be geometrically projected onto a line which is perpendicular to said great circle. It lies within the cylinder archetypes are increased the more in the circumferential direction, the closer they are to the axis lying outside are reduced. To achieve conformality, such a surface element therefore needs to be increased in the axial direction by the same factor. To determine the position in the axial direction, the increase of the distance from the contact line to be projected to the point to be calculated integrated.

Normal Mercator projection

In the normal Mercator projection parallels and meridians are straight lines. Surfaces are stretched by the projection onto the now parallel meridians in east-west direction with the inverse of the cosine of the latitude. In the North-South direction, therefore, the same strain has to be made, the position of a point is then calculated by the integral of the inverse of the cosine of the latitude.

Imaging equations for normal location

The following equations determine the coordinates and a point on a Mercator map from its latitude and longitude ( with the longitude of the map center, angle in radians). The earth is assumed to be spherical; Lengths are made ​​dimensionless with the radius of the earth. The equation for y is the above integral of the reciprocal of the cosine of the latitude ( in place of the tangent at the cylindrical projection gnomonic ):

The inverse is the Gudermann function:

Transverse Mercator projection

The application of the Mercator projection in the transverse position of the imaging cylinder was published in 1772 by Johann Heinrich Lambert. To increase the accuracy Advanced Carl Friedrich Gauss 1825 application to the Erdellipsoid, which was investigated in 1912 by Louis Kruger for the practical application further.

Imaging properties

The most important property of the Mercator projection is its angular loyalty. This also means that in small areas of the length scale in all directions is the same. However, it is constant only along the line of contact and their similarities. Only at contact lines, the projection length is true, that is equivalent to the specified scale. As such it is not equal area. The distortions are a distance from the line of contact progressively larger and infinity at the axis of projection. With two lines of contact area between them is compressed.

The conformality leads in the normal position and the axis loyalty. This means here that the north direction on the map is the same everywhere. Together with the conformality means that rhumb lines (ie constant rates) are shown as straight lines. Shortest connections are not depicted as a straight line, instead, this would be the direction of fidelity required ( → gnomonic Azimuthal ). The closer an area on the north or south pole, the more it is magnified when imaging in normal position. This is the island of Greenland ( 2.2 million km ²) in this map projection presented almost as large as the continent of Africa ( 30.3 million km ²). The North and South Poles can not be displayed because these points lie in the projection at infinity.

Use

The normal Mercator figure is because of their angle and axis loyalty based on almost all charts and some aeronautical charts. The scale changes are noticeable on larger sections of the earth's surface, so that the printed edge of the map Latitude scale is not equidistant. A nautical mile on the chart corresponds exactly to one minute of arc on the same latitude on the left or right edge of the map.

For small-scale maps, especially for the basic maps of many national surveys, the transverse Mercator projection takes the form of Gauss - Kruger projection, Universal transverse Mercator projection ( UTM) and similar large-scale application. The UTM is used in 30 different axial positions for each 6 ° wide strips, Gauss -Krüger used twice as many layers of strips of 3 °. In order to improve the length of fidelity across the wider imaging surface intersects with the UTM projection of the cylinder Erdellipsoid in two lines. In Germany and Austria so far was the Gauss -Krüger projection based on the national survey. The geodesy now necessary international cooperation has helped the internationally -based Gauss - Kruger younger Universal Transverse Mercator projection to enforce, is the converted also in Germany and Austria.

In the Swiss National Survey finds Oblique Mercator projection application in which the axis is chosen so that the fundamental point of the national survey, the observatory in Bern, is located on the same meridian as well as on the touch line.

For large-scale maps, especially maps of the world, insofar as they are not specifically the quotations in the navigation, the Mercator projection is unsuitable because of its rapidly increasing with increasing distance from the contact line distortion. That it was nevertheless used here time and partly resulted from the 1974 to one initiated by the historian Arno Peters discussion about the Mercator projection and the putative alternative known as Gall's orthographic projection in cartography equal-area cylindrical Peters projection. Peters criticized that the Mercator projection a " Euro -centric world view " vermittele because on a world map in the normal Mercator projection countries in relation appear larger in temperate regions such as Europe and other industrialized countries than in the equatorial regions, where lie mainly developing countries.

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